How important is it to have your screenplay registered through the US copyright
office? And if you do get it registered, what happens if you add more scenes
later on?

–Ben Goldblatt

Officially, yes, you should copyright your screenplay (with the little "c" symbol,
name and date) on the title page, and then send it in to the U.S. office, a
procedure you can probably find on-line. And if you make major revisions, you
should probably re-register the whole thing.
Unofficially, nobody does this. Sometimes you’ll see the copyright symbol
on a script, but most of the time you won’t. And none of my writer friends
regularly send in their work to be "officially" copyrighted.

Although it’s not really the same thing, most writers I know do register their
scripts with the Writer’s Guild in Los Angeles, a painless procedure
that can occasionally help if your idea is blatantly stolen. But the truth
is that "someone might steal my idea" is more often the fear of an
aspiring writer who’s never put pen to paper than of a working screenwriter.

I’m ragging on it, but sometimes copyright becomes very important. For instance,
when a script is sold, what the studio is really buying is the copyright. (Or
the right to copyright.) I’m currently adapting BARBARELLA, a project to which
four different studios were claiming copyright. It’s taken the legal teams
more than a year to sort out who really owns what, since two of the original
French comic books were already made into a movie.

The process of determining copyright is called "clearing the chain of
title," and it’s often used as the answer to "Why haven’t they paid
me my money yet?"